75

Chapter 1

−2−4 2 4−2

−4

4

x

y

−2−4 2 4

−4

4

x

y

x

y

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Functions and Their Graphs

1.1 Graphs of Equations1.2 Lines in the Plane1.3 Functions1.4 Graphs of Functions1.5 Shifting, Reflecting, and

Stretching Graphs1.6 Combinations of Functions1.7 Inverse Functions

Selected ApplicationsFunctions have many real-life applications. The applications listedbelow represent a small sample ofthe applications in this chapter.■ Data Analysis,

Exercise 73, page 86■ Rental Demand,

Exercise 86, page 99■ Postal Regulations,

Exercise 81, page 112■ Motor Vehicles,

Exercise 87, page 113■ Fluid Flow,

Exercise 92, page 125■ Finance,

Exercise 58, page 135■ Bacteria,

Exercise 81, page 146■ Consumer Awareness,

Exercises 84, page 146■ Shoe Sizes,

Exercises 103 and 104, page 156

An equation in x and y defines a relationship between the two variables. The equationmay be represented as a graph, providing another perspective on the relationshipbetween x and y. In Chapter 1, you will learn how to write and graph linear equations,how to evaluate and find the domains and ranges of functions, and how to graphfunctions and their transformations.

Refrigeration slows down the activity of bacteria in food so that it takes longer for thebacteria to spoil the food. The number of bacteria in a refrigerated food is a functionof the amount of time the food has been out of refrigeration.

© Index Stock Imagery

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76 Chapter 1 Functions and Their Graphs

Introduction to Library of Parent FunctionsIn Chapter 1, you will be introduced to the concept of a function. As you proceedthrough the text, you will see that functions play a primary role in modeling real-life situations.

There are three basic types of functions that have proven to be the most important in modeling real-life situations. These functions are algebraic functions,exponential and logarithmic functions, and trigonometric and inverse trigonomet-ric functions. These three types of functions are referred to as the elementaryfunctions, though they are often placed in the two categories of algebraic functionsand transcendental functions. Each time a new type of function is studied in detailin this text, it will be highlighted in a box similar to this one. The graphs of manyof these functions are shown on the inside front cover of this text. A review of thesefunctions can be found in the Study Capsules.

Algebraic Functions

These functions are formed by applying algebraic operations to the identity function

Name Function Location

Linear Section 1.2

Quadratic Section 3.1

Cubic Section 3.2

Polynomial Section 3.2

Rational and are polynomial functions Section 3.5

Radical Section 1.3

Transcendental Functions

These functions cannot be formed from the identity function by using algebraic operations.

Name Function Location

Exponential Section 4.1

Logarithmic Section 4.2

Trigonometric

Not covered in this text.

Inverse Trigonometric Not covered in this text.

Nonelementary Functions

Some useful nonelementary functions include the following.

Name Function Location

Absolute value is an elementary function Section 1.3

Piecewise-defined Section 1.3

Greatest integer is an elementary function Section 1.4

Data defined Formula for temperature: Section 1.3F �95

C � 32

g�x�f�x� � �g�x��,

f�x� � � 3x � 2,�2x � 4,

x ≥ 1 x < 1

g�x�f�x� � �g�x��,

f�x� � arcsin x, f�x� � arccos x, f�x� � arctan x

f�x� � csc x, f�x� � sec x, f�x� � cot x

f�x� � sin x, f �x� � cos x, f�x� � tan x,

a � 1a > 0,x > 0,f�x� � loga x,

a � 1a > 0,f�x� � ax,

f�x� � n�P�x�

D�x�N�x�f�x� �N�x�D�x�,

P�x� � anxn � an�1xn�1 � . . . � a2x2 � a1x � a0

f�x� � ax3 � bx2 � cx � d

f�x� � ax2 � bx � c

f�x� � ax � b

f�x� � x.

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